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X-Rays from Proton Bremsstrahlung: Evidence from Fusion Reactors and Its Implication in Astrophysics

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X-Rays from Proton Bremsstrahlung: Evidence from Fusion Reactors and Its Implication in Astrophysics X-rays from Proton Bremsstrahlung: Evidence from Fusion Reactors and Its Implication in Astrophysics Nie Luo Department of Nuclear, Plasma and Radiological Engineering University of Illinois, at Urbana-Champaign Urbana, IL 61801, USA [email protected] ABSTRACT In a fusion reactor, a proton and a neutron generated in previous reactions may again fuse with each other. Or they can in turn fuse with or be captured by an un-reacted deuteron. The average center-of-mass (COM) energy for such reaction is around 10 keV in a typical fusion reactor, but could be as low as 1 keV. At this low COM energy, the reacting nucleons are in an s-wave state in terms of their relative angular momentum. The single- gamma radiation process is thus strongly suppressed due to conservation laws. Instead the gamma ray released is likely to be accompanied by x-ray photons from a nuclear bremsstrahlung process. The x-ray thus generated has a continuous spectrum and peaks around a few hundred eV to a few keV. Therefore, the majority of this nuclear bremsstrahlung radiation is in the form of soft x-ray photons. The average photon energy and spectrum properties of such a process are calculated with a semiclassical approach. The results give a peak near 1.1 keV for the proton-deuteron fusion and a power-to-the-minus-second law in the spectrum’s high-energy limit. The high-energy power law from nuclear bremsstrahlung is harder than that of the ordinary bremsstrahlung from electrons of a Maxwell distribution. The hard x-ray portion of this radiation is therefore not negligible compared to the thermal electron-bremsstrahlung type. An analysis of some prior tokamak discharge data shows that this phenomenon might have been observed before, and its interpretation is complicated by the presence of non-thermal electron bremsstrahlung. Nuclear bremsstrahlung in general and the proton type in particular may lead to new plasma diagnostics which are more sensitive to the ionic or nuclear degree of freedom. Such a prospect is helped by significant radiation in the form of hard x-rays of a few hundred keV. This phenomenon should also play a role in nuclear astrophysics as one of the sources for astrophysical x-rays. The process contributes particularly to stellar evolution in the early stage, where the temperature of proto-stars or the so called pre-main sequence stars (T Tauri stars, for example), is at a relatively low of several million degrees Kelvin. An order-of-magnitude calculation was made on the proton-deuteron fusion rate in young star objects. The estimated x-ray luminosity from this reaction is found enough in magnitude to account for experimental ones. INTRODUCTION For future thermonuclear reactors based on the D-T reaction, the generation of neutrons is the natural consequence of the major power producing reaction. However, protons are also generated by the parasitic D- D reaction according to d + d → 3 H + p , (1) where d is a deuteron. This reaction has a small but non-negligible rate compared to the D-T reaction. In fact its rate is from one to two orders of magnitude within that of D-T between 10 keV and 100 keV. The proton, once generated, could again participate in the thermonuclear process via other reactions. The simplest is the proton capture on neutron, p + n → 2 H + γ . (2) Alternatively, it can fuse with the un-reacted deuterium fuel via p + d → 3 He + γ . (3) Processes in Eqs. (2) and (3) are well known to generate a gamma photon (2.23 MeV and 5.49 MeV respectively), with possible complications to the reactor vessel design. We now consider an associated or parasitic radiation accompanying Eqs. (2) and (3) due to a nuclear bremsstrahlung process, which will generate soft x-rays (one or multiple photons) on the order of one keV besides the MeV gamma. For example, the reaction (3) is more precisely p + d → 3 He + γ + nX , (4) where the total energy carried away by the gamma and the soft x-ray is 5.49 MeV, minus the small recoil kinetic energy of the 3He, which is often negligible. The number of photons is given in n. n =1 should be more likely than the case of multiple photons. However, a clear-cut counting of photon number is difficult in a bremsstrahlung because the radiation is not of a single energy. Similarly reaction (2) is more appropriately p + n → 2 H + γ + nX . (5) The bremsstrahlung x-ray spectrum is continuous, and the photon released in this process is very soft, on the order of a few keV or less. Because soft x-ray of this energy is difficult to detect, and gamma ray in Eq. (4) is still very close to 5.49 MeV, this reaction is difficult to differentiate from the standard one in Eq. (3). In essence, the soft x-ray generation is due to the following physical process. First take the simplest case of p-n capture as an illustration. A proton and a neutron attract each other via the nuclear (strong) force. The strong nuclear attraction causes both nuclei to accelerate to each other. Because the proton is charged, electromagnetic EM radiation is generated according to the theory of classical electrodynamics. The radiation is therefore similar to a bremsstrahlung in the nuclear domain. The release of the energy of the nuclei, is not through the deceleration alone, it also proceeds during acceleration by the strong force because the nucleons undergo both acceleration and deceleration in such a process. Here we loosely term it nuclear Bremsstrahlung, or maybe nuclear Startstrahlung to accentuate its origin in acceleration due to the nuclear force. Such a mechanism radiates x-ray photons around one keV, as we will demonstrate later. The case for proton-deuteron nuclear bremsstrahlung is in principle similar to that of the p-n type. An added complication is due to the Coulomb repulsion. However, once the Coulomb barrier is overcome by quantum mechanical tunneling, the strong nuclear attraction still causes both nuclei to accelerate to each other, albeit at different rates due to the mass difference. Both particles are charged and hence they all radiate EM quanta. Because both particles are positively charged but move in opposite directions, the radiation from them tends to cancel at the far field due to opposite accelerations. The cancellation is however not complete because the acceleration is not identical for proton and deuteron due to their mass difference. The spectrum of such a radiation can be calculated in a semiclassical approach with the inclusion of quantum mechanical effects, as what we will demonstrate in the following section. SEMICLASSICAL RADIATION SPECTRUM The nuclear bremsstrahlung radiation in n-p and p-d capture can be treated semiclassically due to the fact that the energy carried away by the soft photon, on the order of a few hundred eV to a few keV, is much smaller compared to the nuclear potential energy involved and the kinetic energy of the nucleons. This is equivalent to having a Planck constant h approaching 0 so that the quantum mechanical problem returns to a classical one. At the energy and dimensional scales of the problem, the wave nature of the nucleon dominates, and therefore the particle needs to be described by a wavepacket of appreciable spatial extension. However, the radiation sector is adequately handled by a semiclassical treatment of the nucleon acceleration in the nuclear field. The wavepacket nature of the nucleon motion smears out the sharp variation of the μ r potential by convoluting the potential V (r) = V0/(e −1) with the wavefunction ψ of the nucleon, which can be approximated by exponential functions in the coordinate space. The exact functional form of such a convolution is not very important, as we will see, and this type of acceleration in general gives rise to a ω −2 power law in the x-ray spectrum. As long as it is a short range attractive force, the answer is essentially the same. After all, this phenomenon is not a surprise because nuclear scattering at low energy is well known to be potential-shape independent. A detailed semiclassical treatment of the problem in light of quantum mechnical conservation laws is discussed in detail in [1]. Here we briefly quote the similar results derived there, with a straightfoward adoption to the two cases of proton-neutron and proton-deuteron bremsstralung. The high frequency power spectrum is therefore given by e2 4kk 3 1 −2 . (6) P(ω) = − 2 3 2.5 ω 6π ε 0c (a −1) The definition of coefficients is given in [1]. The spectrum at the long wavelength limit is dominated by a ω 3 scaling law for the soft x-ray spectrum when the photon energy approaches 0. The derivation is straightforward and the result is given below e 2 ω 3 (7) P(ω) = 2 3 2 . 6π ε 0 c μ μ in Eq. (7) is given by a specific nuclear-nuclear potential and differs in the two cases of proton-neutron and proton-deuteron radiations. The ω 3 long wavelength limit should intersect the high frequency behavior of Eq. (6) at a frequency ω given by e2 4kk 3 e2 ω 3 − 1 ω −2 = . 6π 2ε c3 (a −1)2.5 6π 2ε c3 μ 2 0 0 (8) Such an ω approximately corresponds to the peak in the bremsstrahlung spectrum ω p . Solving the above equation forω , we obtain an approximate expression for the peak photon frequency 3 1 (2 − a)5 ω ≈ ω = ( )1/5 μk 0.5.
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